CN114970121A - An iterative solution method and system for electromagnetic transient simulation with generator - Google Patents

An iterative solution method and system for electromagnetic transient simulation with generator Download PDF

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CN114970121A
CN114970121A CN202210502080.5A CN202210502080A CN114970121A CN 114970121 A CN114970121 A CN 114970121A CN 202210502080 A CN202210502080 A CN 202210502080A CN 114970121 A CN114970121 A CN 114970121A
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iterative
differential equation
generator
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高路
纪锋
林畅
林俊杰
彭逸轩
庞辉
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State Grid Smart Grid Research Institute of SGCC
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Abstract

An electromagnetic transient simulation iterative solution method and system containing a generator comprises the following steps: establishing a unified nonlinear differential equation for the generator and the circuit system based on electromagnetic transient simulation of the generator; dispersing the unified nonlinear differential equation by adopting an integration method to obtain a dispersed equation; carrying out iterative solution on the dispersed equation by using a Newton iterative method to obtain a solution in each time step; according to the invention, through the integral differential equation of the column writing power system, the nonlinear equation is solved by using the Newton Raphson method, so that iterative solution of the motor and the circuit system in a time step can be realized, and the numerical divergence problem caused by separate solution delay of the motor and the circuit system or the numerical stability and precision problem caused by inaccurate prediction method are solved.

Description

一种含发电机的电磁暂态仿真迭代求解方法及系统An iterative solution method and system for electromagnetic transient simulation with generator

技术领域technical field

本发明涉及电力系统电磁暂态仿真领域,具体涉及一种含发电机的电磁暂态仿真迭代求解方法及系统。The invention relates to the field of electromagnetic transient simulation of power systems, in particular to an iterative solution method and system for electromagnetic transient simulation including a generator.

背景技术Background technique

在现有的电磁暂态仿真中,发电机等非线性元件通常被等效为受控源与主电路连接计算,通常采用两种方式对受控源的值进行更新。1、需要从已知t-Δt时刻的解预测一些t时刻尚未知的量,结果准确性较依赖于采用的预测方法,预报方法选取的不合适将导致数值不稳定;2、采用已知的t-Δt时刻的量计算,可能存在由于延时造成的数值发散问题,需对延时进行补偿校正。上述两种方式都存在数值稳定性问题,只有将求解方法从直接求解改变成在每一时步内的迭代解,其数值稳定性才能进一步提高。In the existing electromagnetic transient simulation, nonlinear components such as generators are usually calculated equivalently as the controlled source connected to the main circuit, and the value of the controlled source is usually updated in two ways. 1. It is necessary to predict some quantities that are not known at time t from the solution at time t-Δt. The accuracy of the results depends on the prediction method used. The inappropriate selection of the prediction method will lead to numerical instability; 2. Use the known There may be a problem of numerical divergence caused by the delay in the calculation of the quantity at the time t-Δt, and the delay needs to be compensated and corrected. The above two methods have numerical stability problems. Only by changing the solution method from direct solution to iterative solution in each time step, the numerical stability can be further improved.

发明内容SUMMARY OF THE INVENTION

为了解决现有技术中发电机电磁暂态仿真存在的数值稳定性问题,本发明提出了一种含发电机的电磁暂态仿真迭代求解方法,包括:In order to solve the numerical stability problem existing in the electromagnetic transient simulation of generators in the prior art, the present invention proposes an iterative solution method for electromagnetic transient simulation with generators, including:

基于发电机的电磁暂态仿真将发电机和电路系统建立统一的非线性微分方程;The generator-based electromagnetic transient simulation establishes a unified nonlinear differential equation for the generator and circuit system;

采用积分方法对所述统一的非线性微分方程进行离散,得到离散后的方程;The unified nonlinear differential equation is discretized by the integral method to obtain the discretized equation;

利用牛顿迭代法对所述离散后的方程进行迭代求解,得到每个时间步长内的解。The discrete equations are iteratively solved by using the Newton iteration method, and the solutions in each time step are obtained.

优选的,所述采用积分方法对所述统一的非线性微分方程进行离散,得到离散后的方程,包括:Preferably, the integral method is used to discretize the unified nonlinear differential equation to obtain a discretized equation, including:

将所述统一的非线性微分方程转换为一阶非线性微分方程;converting the unified nonlinear differential equation into a first-order nonlinear differential equation;

利用不同的积分方法对所述一阶非线性微分方程进行离散,得到离散后的方程。The first-order nonlinear differential equation is discretized by using different integration methods to obtain the discretized equation.

优选的,所述利用不同的积分方法对所述一阶非线性微分方程进行离散,得到离散后的方程,包括:Preferably, different integration methods are used to discretize the first-order nonlinear differential equation to obtain a discretized equation, including:

将在t时刻对所述一阶非线性微分方程乘以1-β得到的公式与在t+Δt时刻对所述一阶非线性微分方程乘以β得到的公式相加,得到由t时刻计算t+Δt 时刻的计算式;Adding the formula obtained by multiplying the first-order nonlinear differential equation by 1-β at time t and the formula obtained by multiplying the first-order nonlinear differential equation by β at time t+Δt, the calculation is obtained at time t. Calculation formula at time t+Δt;

将所述在t时刻对所述一非线性阶微分方程乘以1-β得到的公式乘以

Figure BDA0003634755160000021
后,减去所述由t时刻计算t+Δt时刻的计算式,得到离散后的方程。Multiply the formula obtained by multiplying the first nonlinear differential equation by 1-β at time t by
Figure BDA0003634755160000021
Then, subtract the calculation formula for calculating time t+Δt from time t to obtain a discrete equation.

优选的,所述统一的非线性微分方程的计算式如下:Preferably, the calculation formula of the unified nonlinear differential equation is as follows:

Figure BDA0003634755160000022
Figure BDA0003634755160000022

式中,n为电路节点个数,nm为电机机械轴系质量块数,L(θ)为发电机电感矩阵,B为发电机绕组和节点的关联矩阵,J为发电机轴系转动惯性矩阵,D为阻尼矩阵,K为弹性系数矩阵,T为发电机机械转矩向量,KC为电容系数矩阵, KR为电阻系数矩阵,KL为电感系数矩阵,Ψ为节点磁链,θ为电机轴系转角。In the formula, n is the number of circuit nodes, nm is the number of mass blocks in the mechanical shaft system of the motor, L(θ) is the generator inductance matrix, B is the correlation matrix between the generator windings and nodes, and J is the rotational inertia matrix of the generator shaft system , D is the damping matrix, K is the elastic coefficient matrix, T is the generator mechanical torque vector, K C is the capacitance coefficient matrix, K R is the resistance coefficient matrix, K L is the inductance coefficient matrix, Ψ is the node flux linkage, θ is Motor shafting angle.

优选的,所述利用牛顿迭代法对所述离散后的方程进行迭代求解,得到每个时间步长内的解,包括:Preferably, the discrete equation is iteratively solved by the Newton iteration method to obtain a solution within each time step, including:

将所述离散后的方程转换为函数式,并获取函数式的雅克比矩阵;converting the discretized equation into a functional formula, and obtaining a Jacobian matrix of the functional formula;

将所述雅克比矩阵转换为可以利用牛顿迭代法的迭代式;transforming the Jacobian matrix into an iterative formula that can utilize Newton's iterative method;

利用牛顿迭代法对所述迭代式进行迭代求解,得到每个时间步长内的解。The iterative formula is iteratively solved by using the Newton iteration method, and the solution in each time step is obtained.

优选的,所述利用牛顿迭代法对所述迭代式进行迭代求解,得到每个时间步长内的解,包括:Preferably, the iterative solution is iteratively solved by using the Newton iteration method to obtain a solution within each time step, including:

将‖f(xN+1,k+1)‖与设定的误差值进行比较,若小于所述设定的误差值则计算结束,否则按照牛顿迭代法的迭代式进行求解,xN+1,k+1为第k+1次迭代的估计值,k为迭代次数,x为状态量,f(xN+1,k+1)为第N+1步计算的第k+1次迭代的函数值,N为第N步计算。Compare ‖f(x N+1,k+1 )‖ with the set error value, if it is less than the set error value, the calculation ends, otherwise, solve according to the iterative formula of Newton iteration method, x N+ 1,k+1 is the estimated value of the k+1th iteration, k is the number of iterations, x is the state quantity, and f(x N+1,k+1 ) is the k+1th time calculated in the N+1th step The function value of the iteration, N is the calculation of the Nth step.

优选的,所述利用牛顿迭代法的迭代式如下:Preferably, the iterative formula using the Newton iteration method is as follows:

Figure BDA0003634755160000023
Figure BDA0003634755160000023

式中,xN+1,k为t+Δt时刻初始的估计值,xN+1,k+1为新的结果估计值,即下一次迭代初始的估计值。In the formula, x N+1, k is the initial estimated value at time t+Δt, and x N+1, k+1 is the new result estimated value, that is, the initial estimated value of the next iteration.

基于同一发明构思,本发明还提出了一种含发电机的电磁暂态仿真迭代求解系统,包括:Based on the same inventive concept, the present invention also proposes an electromagnetic transient simulation iterative solution system including a generator, including:

微分方程建立模块,用于基于发电机的电磁暂态仿真将发电机和电路系统建立统一的非线性微分方程;The differential equation building module is used to establish a unified nonlinear differential equation for the generator and the circuit system based on the electromagnetic transient simulation of the generator;

微分方程离散模块,用于采用积分方法对所述统一的非线性微分方程进行离散,得到离散后的方程;a differential equation discretization module, used for discretizing the unified nonlinear differential equation by using the integral method to obtain the discretized equation;

迭代求解模块,用于利用牛顿迭代法对所述离散后的方程进行迭代求解,得到每个时步内的解。The iterative solution module is used to iteratively solve the discretized equation by using the Newton iteration method to obtain the solution in each time step.

优选的,所述微分方程离散模块,包括:Preferably, the differential equation discrete module includes:

转换子模块,用于将所述统一的非线性微分方程转换为一阶非线性微分方程;a conversion submodule for converting the unified nonlinear differential equation into a first-order nonlinear differential equation;

离散子模块,用于利用不同的积分方法对所述一阶非线性微分方程进行离散,得到离散后的方程。The discrete sub-module is used for discretizing the first-order nonlinear differential equation by using different integration methods to obtain the discretized equation.

优选的,所述离散子模块,具体用于:Preferably, the discrete submodule is specifically used for:

将在t时刻对所述一阶非线性微分方程乘以1-β得到的公式与在t+Δt时刻对所述一阶非线性微分方程乘以β得到的公式相加,得到由t时刻计算t+Δt 时刻的计算式;Adding the formula obtained by multiplying the first-order nonlinear differential equation by 1-β at time t and the formula obtained by multiplying the first-order nonlinear differential equation by β at time t+Δt, the calculation is obtained at time t. Calculation formula at time t+Δt;

将所述在t时刻对所述一阶非线性微分方程乘以1-β得到的公式乘以

Figure BDA0003634755160000031
后,减去所述由t时刻计算t+Δt时刻的计算式,得到离散后的方程。Multiply the formula obtained by multiplying the first-order nonlinear differential equation by 1-β at time t by
Figure BDA0003634755160000031
Then, subtract the calculation formula for calculating time t+Δt from time t to obtain a discrete equation.

优选的,所述统一的非线性微分方程的计算式如下:Preferably, the calculation formula of the unified nonlinear differential equation is as follows:

Figure BDA0003634755160000032
Figure BDA0003634755160000032

式中,n为电路节点个数,nm为电机机械轴系质量块数,L(θ)为发电机电感矩阵,B为发电机绕组和节点的关联矩阵,J为发电机轴系转动惯性矩阵,D为阻尼矩阵,K为弹性系数矩阵,T为发电机机械转矩向量,KC为电容系数矩阵, KR为电阻系数矩阵,KL为电感系数矩阵,Ψ为节点磁链,θ为电机轴系转角。In the formula, n is the number of circuit nodes, nm is the number of mass blocks in the mechanical shaft system of the motor, L(θ) is the generator inductance matrix, B is the correlation matrix between the generator windings and nodes, and J is the rotational inertia matrix of the generator shaft system , D is the damping matrix, K is the elastic coefficient matrix, T is the generator mechanical torque vector, K C is the capacitance coefficient matrix, K R is the resistance coefficient matrix, K L is the inductance coefficient matrix, Ψ is the node flux linkage, θ is Motor shafting angle.

优选的,所述迭代求解模块,包括:Preferably, the iterative solution module includes:

函数式子模块,用于将所述离散后的方程转换为函数式,并获取函数式的雅克比矩阵;a functional submodule, used to convert the discrete equation into a functional formula, and obtain a Jacobian matrix of the functional formula;

迭代式子模块,用于将所述雅克比矩阵转换为可以利用牛顿迭代法的迭代式;an iterative submodule for converting the Jacobian matrix into an iterative formula that can utilize Newton's iterative method;

求解子模块,用于利用牛顿迭代法对所述迭代式进行迭代求解,得到每个时间步长内的解。The solving submodule is used to iteratively solve the iterative formula by using the Newton iterative method to obtain the solution in each time step.

优选的,所述求解子模块具体用于:Preferably, the solving submodule is specifically used for:

将‖f(xN+1,k+1)‖与设定的误差值进行比较,若小于所述设定的误差值则计算结束,否则按照牛顿迭代法的迭代式进行求解,xN+1,k+1为第k+1次迭代的估计值,k为迭代次数,x为状态量,f(xN+1,k+1)为第N+1步计算的第k+1次迭代的函数值,N为第N步计算。Compare ‖f(x N+1,k+1 )‖ with the set error value, if it is less than the set error value, the calculation ends, otherwise, solve according to the iterative formula of Newton iteration method, x N+ 1,k+1 is the estimated value of the k+1th iteration, k is the number of iterations, x is the state quantity, and f(x N+1,k+1 ) is the k+1th time calculated in the N+1th step The function value of the iteration, N is the calculation of the Nth step.

优选的,所述利用牛顿迭代法的迭代式如下:Preferably, the iterative formula using the Newton iteration method is as follows:

Figure BDA0003634755160000041
Figure BDA0003634755160000041

式中,xN+1k为t+Δt时刻初始的估计值,xN+1,k+1为新的结果估计值,即下一次迭代初始的估计值。In the formula, x N+1 , k is the initial estimated value at time t+Δt, and x N+1, k+1 is the new result estimated value, that is, the initial estimated value of the next iteration.

与现有技术相比,本发明的有益效果为:Compared with the prior art, the beneficial effects of the present invention are:

一种含发电机的电磁暂态仿真迭代求解方法及系统,包括:基于发电机的电磁暂态仿真将发电机和电路系统建立统一的非线性微分方程;采用积分方法对所述统一的非线性微分方程进行离散,得到离散后的方程;利用牛顿迭代法对所述离散后的方程进行迭代求解,得到每个时间步长内的解;本发明通过列写电力系统的整体微分方程,利用牛顿拉夫逊法求解非线性方程,可实现电机和电路系统在一个时间步长内的迭代求解,从而解决由电机和电路系统分开求解延时造成的数值发散问题或由预测方法不精确造成的数值稳定性和精度问题。An iterative solution method and system for electromagnetic transient simulation including a generator, comprising: establishing a unified nonlinear differential equation for a generator and a circuit system based on the electromagnetic transient simulation of the generator; The differential equation is discretized to obtain the discrete equation; the discrete equation is iteratively solved by the Newton iteration method to obtain the solution in each time step; the present invention writes the overall differential equation of the power system by using Newton’s iterative solution. The Raphson method solves nonlinear equations, which can realize the iterative solution of the motor and circuit system within one time step, so as to solve the numerical divergence problem caused by the separate solution delay of the motor and the circuit system or the numerical stability caused by the inaccurate prediction method. performance and precision issues.

附图说明Description of drawings

图1为本发明的一种含发电机的电磁暂态仿真迭代求解方法流程图;Fig. 1 is a kind of electromagnetic transient simulation iterative solution method flow chart containing generator of the present invention;

图2为本发明的每个时步迭代求解流程图。FIG. 2 is a flowchart of iterative solution for each time step of the present invention.

具体实施方式Detailed ways

本发明提出一种含发电机电磁暂态仿真迭代求解方法,针对发电机电磁暂态仿真等效为受控源与电路连接存在的数值稳定性问题,通过将发电机和电路系统建立统一的微分方程,使用牛顿拉夫逊迭代法对整体方程进行迭代求解,从根本上避免发电机被等效为受控源由于延时或预测方法带来的数值稳定性问题。The invention proposes an iterative solution method for electromagnetic transient simulation involving generators, aiming at the numerical stability problem that the electromagnetic transient simulation of generators is equivalent to the connection between controlled sources and circuits, by establishing a unified differential between generators and circuit systems The overall equation is solved iteratively using the Newton-Raphson iteration method, which fundamentally avoids the numerical stability problem caused by the generator being equivalent to a controlled source due to delay or prediction methods.

实施例1:Example 1:

一种含发电机的电磁暂态仿真迭代求解方法,具体过程如图1所示,包括:An iterative solution method for electromagnetic transient simulation with generator, the specific process is shown in Figure 1, including:

步骤1,基于发电机的电磁暂态仿真将发电机和电路系统建立统一的非线性微分方程;Step 1, establish a unified nonlinear differential equation for the generator and the circuit system based on the electromagnetic transient simulation of the generator;

步骤2,采用积分方法对所述统一的非线性微分方程进行离散,得到离散后的方程;Step 2, using the integral method to discretize the unified nonlinear differential equation to obtain the discretized equation;

步骤3,利用牛顿迭代法对所述离散后的方程进行迭代求解,得到每个时间步长内的解。Step 3, iteratively solve the discretized equation by using the Newton iteration method, and obtain the solution in each time step.

下面对本发明的一种含发电机的电磁暂态仿真迭代求解方法,结合图2进行详细介绍。An iterative solution method for electromagnetic transient simulation including a generator of the present invention is described in detail below with reference to FIG. 2 .

步骤1中的,基于发电机的电磁暂态仿真将发电机和电路系统建立统一的非线性微分方程,具体包括:In step 1, a unified nonlinear differential equation is established for the generator and the circuit system based on the electromagnetic transient simulation of the generator, which specifically includes:

列写的电力系统在时域求解中的微分方程:Write the differential equations of the power system in the time domain solution:

Figure BDA0003634755160000051
Figure BDA0003634755160000051

式中,KC为电容系数矩阵,KR为电阻系数矩阵,KL为电感系数矩阵,Ψ为节点磁链。In the formula, K C is the capacitance coefficient matrix, K R is the resistance coefficient matrix, K L is the inductance coefficient matrix, and Ψ is the node flux linkage.

扩展列写含发电机的微分方程:Expand the column to write a differential equation with generators:

Figure BDA0003634755160000052
Figure BDA0003634755160000052

其中,n为电路节点个数,nm为电机机械轴系质量块数。L(θ)为发电机电感矩阵,是随发电机转角变化的,B为发电机绕组和节点的关联矩阵。J为发电机轴系转动惯量矩阵,D为阻尼矩阵,K为弹性系数矩阵,T为发电机机械转矩向量。Among them, n is the number of circuit nodes, and nm is the number of mass blocks in the mechanical shaft system of the motor. L(θ) is the inductance matrix of the generator, which changes with the rotation angle of the generator, and B is the correlation matrix of the generator windings and nodes. J is the rotational inertia matrix of the generator shaft system, D is the damping matrix, K is the elastic coefficient matrix, and T is the generator mechanical torque vector.

步骤2中的,采用积分方法对所述统一的非线性微分方程进行离散,得到离散后的方程,具体包括:In step 2, the integral method is used to discretize the unified nonlinear differential equation to obtain a discretized equation, which specifically includes:

电力系统的电磁暂态仿真就是求解式(2)所示的微分方程。The electromagnetic transient simulation of the power system is to solve the differential equation shown in equation (2).

式(2)是一个二阶微分方程,需变换为一阶微分方程进行计算机求解。将式(2)变换为:Equation (2) is a second-order differential equation, which needs to be transformed into a first-order differential equation for computer solution. Transform equation (2) into:

Figure BDA0003634755160000061
Figure BDA0003634755160000061

其中,

Figure BDA0003634755160000062
Figure BDA0003634755160000063
E为单位矩阵,
Figure BDA0003634755160000064
Figure BDA0003634755160000065
in,
Figure BDA0003634755160000062
Figure BDA0003634755160000063
E is the identity matrix,
Figure BDA0003634755160000064
Figure BDA0003634755160000065

利用不同积分方法对式(3)离散:Equation (3) is discretized using different integration methods:

Figure BDA0003634755160000066
Figure BDA0003634755160000066

其中,Δt为离散时间步长,xN为t时刻的状态量,xN+1为t+Δt时刻的状态量,β为不同积分方法的选择因子。当β=0.5时,为梯形积分法;β=1时,为后退欧拉法。Among them, Δt is the discrete time step, xN is the state quantity at time t, xN+1 is the state quantity at time t+Δt, and β is the selection factor of different integration methods. When β=0.5, it is the trapezoidal integration method; when β=1, it is the backward Euler method.

在t时刻,对式(3)乘以1-β,得:At time t, multiply Equation (3) by 1-β to get:

Figure BDA0003634755160000067
Figure BDA0003634755160000067

在t+Δt时刻,对式(3)乘以β,得:At time t+Δt, multiply the equation (3) by β to get:

Figure BDA0003634755160000068
Figure BDA0003634755160000068

由t时刻计算t+Δt时刻,K1和K2保持不变,式(5)和式(6)相加,得:Calculate time t+Δt from time t, K 1 and K 2 remain unchanged, and formula (5) and formula (6) are added to obtain:

Figure BDA0003634755160000069
Figure BDA0003634755160000069

将式(5)乘以

Figure BDA00036347551600000610
得:Multiply equation (5) by
Figure BDA00036347551600000610
have to:

Figure BDA0003634755160000071
Figure BDA0003634755160000071

式(8)减去式(7)得:Formula (8) is subtracted from formula (7) to get:

Figure BDA0003634755160000072
Figure BDA0003634755160000072

进一步化简式(9)得:Further simplifying equation (9), we get:

AxN+1=BxN+(1-β)R(xN)+βR(xN+1) (10)Ax N+1 =Bx N +(1-β)R(x N )+βR(x N+1 ) (10)

其中,

Figure BDA0003634755160000073
in,
Figure BDA0003634755160000073

由于式(10)中R(xN)和R(xN+1)是关于xN和xN+1的非线性函数,故式(10) 为非线性方程。Since R(x N ) and R(x N+1 ) in equation (10) are nonlinear functions about x N and x N+1 , equation (10) is a nonlinear equation.

步骤3中的,利用牛顿迭代法对所述离散后的方程进行迭代求解,得到每个时间步长内的解,具体包括:In step 3, the discrete equation is iteratively solved by the Newton iteration method, and the solution in each time step is obtained, which specifically includes:

式(10)中的未知变量是xN+1,而xN和R(xN)均为已知量。为了使用牛顿迭代法求解,需将式(10)变成式(11)的形式,即为求解f(xN+1)=0。The unknown variable in equation (10) is x N+1 , and both x N and R(x N ) are known quantities. In order to use the Newton iteration method to solve, it is necessary to change the formula (10) into the form of the formula (11), that is, to solve f(x N+1 )=0.

f(xN+1)=-AxN+1+BxN+(1-β)R(xN)+βR(xN+1)=0 (11)f(x N+1 )=-Ax N+1 +Bx N +(1-β)R(x N )+βR(x N+1 )=0 (11)

函数f(xN+1)的雅克比矩阵为:The Jacobian matrix of the function f(x N+1 ) is:

Figure BDA0003634755160000074
Figure BDA0003634755160000074

其中,in,

Figure BDA0003634755160000075
Γ(θ)= (L(θ))-1
Figure BDA0003634755160000076
Figure BDA0003634755160000077
都是关于θ的矩阵函数,可根据电机类型及参数具体确定。
Figure BDA0003634755160000075
Γ(θ) = (L(θ)) -1 ,
Figure BDA0003634755160000076
and
Figure BDA0003634755160000077
They are all matrix functions about θ, which can be determined according to the motor type and parameters.

利用牛顿迭代法的迭代式为:The iterative formula using Newton's iterative method is:

Figure BDA0003634755160000078
Figure BDA0003634755160000078

其中xN+1,k是t+Δt时刻初始的估计值,xN+1,k+1是新的结果估计值,即下一次迭代初始的估计值。当‖f(xN+1,k+1)‖小于设定允许误差时,则完成每个时步内的迭代求解。Where x N+1, k is the initial estimated value at time t+Δt, and x N+1, k+1 is the new result estimated value, that is, the initial estimated value of the next iteration. When ‖f(x N+1,k+1 )‖ is less than the set allowable error, the iterative solution in each time step is completed.

实施例2:Example 2:

一种含发电机的电磁暂态仿真迭代求解系统,包括:An iterative solution system for electromagnetic transient simulation with generator, comprising:

微分方程建立模块,用于基于发电机的电磁暂态仿真将发电机和电路系统建立统一的非线性微分方程;The differential equation building module is used to establish a unified nonlinear differential equation for the generator and the circuit system based on the electromagnetic transient simulation of the generator;

微分方程离散模块,用于采用积分方法对所述统一的非线性微分方程进行离散,得到离散后的方程;a differential equation discretization module, used for discretizing the unified nonlinear differential equation by using the integral method to obtain the discretized equation;

迭代求解模块,用于利用牛顿迭代法对所述离散后的方程进行迭代求解,得到每个时步内的解。The iterative solution module is used to iteratively solve the discretized equation by using the Newton iteration method to obtain the solution in each time step.

所述微分方程离散模块,包括:The differential equation discrete module includes:

转换子模块,用于将所述统一的非线性微分方程转换为一阶非线性微分方程;a conversion submodule for converting the unified nonlinear differential equation into a first-order nonlinear differential equation;

离散子模块,用于利用不同的积分方法对所述一阶非线性微分方程进行离散得到离散后的方程。The discrete sub-module is used for discretizing the first-order nonlinear differential equation by using different integration methods to obtain a discretized equation.

所述迭代求解模块,包括:The iterative solution module includes:

函数式子模块,用于将所述离散后的方程转换为函数式,并获取函数式的雅克比矩阵;a functional submodule, used to convert the discrete equation into a functional formula, and obtain a Jacobian matrix of the functional formula;

迭代式子模块,用于将所述雅克比矩阵转换为可以利用牛顿迭代法的迭代式;an iterative submodule for converting the Jacobian matrix into an iterative formula that can utilize Newton's iterative method;

求解子模块,用于利用牛顿迭代法对所述迭代式进行迭代求解,得到每个时间步长内的解。The solving submodule is used to iteratively solve the iterative formula by using the Newton iterative method to obtain the solution in each time step.

微分方程建立模块,具体用于:Differential equation building block, specifically for:

列写的电力系统在时域求解中的微分方程:Write the differential equations of the power system in the time domain solution:

Figure BDA0003634755160000081
Figure BDA0003634755160000081

式中,KC为电容系数矩阵,KR为电阻系数矩阵,KL为电感系数矩阵,Ψ为节点磁链。In the formula, K C is the capacitance coefficient matrix, K R is the resistance coefficient matrix, K L is the inductance coefficient matrix, and Ψ is the node flux linkage.

扩展列写含发电机的微分方程:Expand the column to write a differential equation with generators:

Figure BDA0003634755160000082
Figure BDA0003634755160000082

其中,n为电路节点个数,nm为电机机械轴系质量块数。L(θ)为发电机电感矩阵,是随发电机转角变化的,B为发电机绕组和节点的关联矩阵。J为发电机轴系转动惯量矩阵,D为阻尼矩阵,K为弹性系数矩阵,T为发电机机械转矩向量。Among them, n is the number of circuit nodes, and nm is the number of mass blocks in the mechanical shaft system of the motor. L(θ) is the inductance matrix of the generator, which changes with the rotation angle of the generator, and B is the correlation matrix of the generator windings and nodes. J is the rotational inertia matrix of the generator shaft system, D is the damping matrix, K is the elastic coefficient matrix, and T is the generator mechanical torque vector.

转换子模块,具体用于:Transform submodules, specifically for:

电力系统的电磁暂态仿真就是求解式(2)所示的微分方程。The electromagnetic transient simulation of the power system is to solve the differential equation shown in equation (2).

式(2)是一个二阶微分方程,需变换为一阶微分方程进行计算机求解。将式(2)变换为:Equation (2) is a second-order differential equation, which needs to be transformed into a first-order differential equation for computer solution. Transform equation (2) into:

Figure BDA0003634755160000091
Figure BDA0003634755160000091

其中,

Figure BDA0003634755160000092
Figure BDA0003634755160000093
E为单位矩阵,
Figure BDA0003634755160000094
Figure BDA0003634755160000095
in,
Figure BDA0003634755160000092
Figure BDA0003634755160000093
E is the identity matrix,
Figure BDA0003634755160000094
Figure BDA0003634755160000095

离散子模块,具体用于:Discrete submodules, specifically for:

利用不同积分方法对式(3)离散:Equation (3) is discretized using different integration methods:

Figure BDA0003634755160000096
Figure BDA0003634755160000096

其中,Δt为离散时间步长,xN为t时刻的状态量,xN+1为t+Δt时刻的状态量,β为不同积分方法的选择因子。当β=0.5时,为梯形积分法;β=1时,为后退欧拉法。Among them, Δt is the discrete time step, xN is the state quantity at time t, xN+1 is the state quantity at time t+Δt, and β is the selection factor of different integration methods. When β=0.5, it is the trapezoidal integration method; when β=1, it is the backward Euler method.

在t时刻,对式(3)乘以1-β,得:At time t, multiply Equation (3) by 1-β to get:

Figure BDA0003634755160000097
Figure BDA0003634755160000097

在t+Δt时刻,对式(3)乘以β,得:At time t+Δt, multiply the equation (3) by β to get:

Figure BDA0003634755160000101
Figure BDA0003634755160000101

由t时刻计算t+Δt时刻,K1和K2保持不变,式(5)和式(6)相加,得:Calculate time t+Δt from time t, K 1 and K 2 remain unchanged, and formula (5) and formula (6) are added to obtain:

Figure BDA0003634755160000102
Figure BDA0003634755160000102

将式(5)乘以

Figure BDA0003634755160000103
得:Multiply equation (5) by
Figure BDA0003634755160000103
have to:

Figure BDA0003634755160000104
Figure BDA0003634755160000104

式(8)减去式(7)得:Formula (8) is subtracted from formula (7) to get:

Figure BDA0003634755160000105
Figure BDA0003634755160000105

进一步化简式(9)得:Further simplifying equation (9), we get:

AxN+1=BxN+(1-β)R(xN)+βR(xN+1) (10)Ax N+1 =Bx N +(1-β)R(x N )+βR(x N+1 ) (10)

其中,

Figure BDA0003634755160000106
in,
Figure BDA0003634755160000106

由于式(10)中R(xN)和R(xN+1)是关于xN和xN+1的非线性函数,故式(10) 为非线性方程。Since R(x N ) and R(x N+1 ) in equation (10) are nonlinear functions about x N and x N+1 , equation (10) is a nonlinear equation.

函数式子模块,具体用于:Functional submodules for:

式(10)中的未知变量是xN+1,而xN和R(xN)均为已知量。为了使用牛顿迭代法求解,需将式(10)变成式(11)的形式,即为求解f(xN+1)=0。The unknown variable in equation (10) is x N+1 , and both x N and R(x N ) are known quantities. In order to use the Newton iteration method to solve, it is necessary to change the formula (10) into the form of the formula (11), that is, to solve f(x N+1 )=0.

f(xN+1)=-AxN+1+BxN+(1-β)R(xN)+βR(xN+1)=0 (11)f(x N+1 )=-Ax N+1 +Bx N +(1-β)R(x N )+βR(x N+1 )=0 (11)

函数f(xN+1)的雅克比矩阵为:The Jacobian matrix of the function f(x N+1 ) is:

Figure BDA0003634755160000107
Figure BDA0003634755160000107

其中,in,

Figure BDA0003634755160000108
Γ(θ)= (L(θ))-1
Figure BDA0003634755160000109
Figure BDA00036347551600001010
都是关于θ的矩阵函数,可根据电机类型及参数具体确定。
Figure BDA0003634755160000108
Γ(θ) = (L(θ)) -1 ,
Figure BDA0003634755160000109
and
Figure BDA00036347551600001010
They are all matrix functions about θ, which can be determined according to the motor type and parameters.

迭代式子模块,具体用于:Iterative submodule for:

利用牛顿迭代法的迭代式为:The iterative formula using Newton's iterative method is:

Figure BDA0003634755160000111
Figure BDA0003634755160000111

其中xN+1,k是t+Δt时刻初始的估计值,xN+1,k+1是新的结果估计值,即下一次迭代初始的估计值。Where x N+1, k is the initial estimated value at time t+Δt, and x N+1, k+1 is the new result estimated value, that is, the initial estimated value of the next iteration.

求解子模块,具体用于:Solve submodules for:

当‖f(xN+1,k+1)‖小于设定允许误差时,则完成每个时步内的迭代求解。When ‖f(x N+1,k+1 )‖ is less than the set allowable error, the iterative solution in each time step is completed.

本领域内的技术人员应明白,本发明的实施例可提供为方法、系统、或计算机程序产品。因此,本发明可采用完全硬件实施例、完全软件实施例、或结合软件和硬件方面的实施例的形式。而且,本发明可采用在一个或多个其中包含有计算机可用程序代码的计算机可用存储介质(包括但不限于磁盘存储器、CD-ROM、光学存储器等)上实施的计算机程序产品的形式。As will be appreciated by one skilled in the art, embodiments of the present invention may be provided as a method, system, or computer program product. Accordingly, the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) having computer-usable program code embodied therein.

本发明是参照根据本发明实施例的方法、设备(系统)、和计算机程序产品的流程图和/或方框图来描述的。应理解可由计算机程序指令实现流程图和/或方框图中的每一流程和/或方框、以及流程图和/或方框图中的流程和/或方框的结合。可提供这些计算机程序指令到通用计算机、专用计算机、嵌入式处理机或其他可编程数据处理设备的处理器以产生一个机器,使得通过计算机或其他可编程数据处理设备的处理器执行的指令产生用于实现在流程图一个流程或多个流程和/或方框图一个方框或多个方框中指定的功能的装置。The present invention is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each flow and/or block in the flowchart illustrations and/or block diagrams, and combinations of flows and/or blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to the processor of a general purpose computer, special purpose computer, embedded processor or other programmable data processing device to produce a machine such that the instructions executed by the processor of the computer or other programmable data processing device produce Means for implementing the functions specified in a flow or flow of a flowchart and/or a block or blocks of a block diagram.

这些计算机程序指令也可存储在能引导计算机或其他可编程数据处理设备以特定方式工作的计算机可读存储器中,使得存储在该计算机可读存储器中的指令产生包括指令装置的制造品,该指令装置实现在流程图一个流程或多个流程和 /或方框图一个方框或多个方框中指定的功能。These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory result in an article of manufacture comprising instruction means, the instructions The apparatus implements the functions specified in the flow or flow of the flowcharts and/or the block or blocks of the block diagrams.

这些计算机程序指令也可装载到计算机或其他可编程数据处理设备上,使得在计算机或其他可编程设备上执行一系列操作步骤以产生计算机实现的处理,从而在计算机或其他可编程设备上执行的指令提供用于实现在流程图一个流程或多个流程和/或方框图一个方框或多个方框中指定的功能的步骤。These computer program instructions can also be loaded on a computer or other programmable data processing device to cause a series of operational steps to be performed on the computer or other programmable device to produce a computer-implemented process such that The instructions provide steps for implementing the functions specified in the flow or blocks of the flowcharts and/or the block or blocks of the block diagrams.

以上仅为本发明的实施例而已,并不用于限制本发明,凡在本发明的精神和原则之内,所做的任何修改、等同替换、改进等,均包含在发明待批的本发明的权利要求范围之内。The above are only embodiments of the present invention, and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention are included in the invention pending approval. within the scope of the claims.

Claims (14)

1.一种含发电机的电磁暂态仿真迭代求解方法,其特征在于,包括:1. a kind of electromagnetic transient simulation iterative solution method containing generator, is characterized in that, comprises: 基于发电机的电磁暂态仿真将发电机和电路系统建立统一的非线性微分方程;The generator-based electromagnetic transient simulation establishes a unified nonlinear differential equation for the generator and circuit system; 采用积分方法对所述统一的非线性微分方程进行离散,得到离散后的方程;The unified nonlinear differential equation is discretized by the integral method to obtain the discretized equation; 利用牛顿迭代法对所述离散后的方程进行迭代求解,得到每个时间步长内的解。The discrete equations are iteratively solved by using the Newton iteration method, and the solutions in each time step are obtained. 2.根据权利要求1所述一种含发电机的电磁暂态仿真迭代求解方法,其特征在于,所述采用积分方法对所述统一的非线性微分方程进行离散,得到离散后的方程,包括:2. A kind of electromagnetic transient simulation iterative solution method containing generator according to claim 1, is characterized in that, described adopting integral method to discretize described unified nonlinear differential equation, obtain the equation after discretization, comprising : 将所述统一的非线性微分方程转换为一阶非线性微分方程;converting the unified nonlinear differential equation into a first-order nonlinear differential equation; 利用不同的积分方法对所述一阶非线性微分方程进行离散得到离散后的方程。Different integration methods are used to discretize the first-order nonlinear differential equation to obtain a discretized equation. 3.根据权利要求2所述一种含发电机的电磁暂态仿真迭代求解方法,其特征在于,所述利用不同的积分方法对所述一阶非线性微分方程进行离散得到离散后的方程,包括:3. a kind of electromagnetic transient simulation iterative solution method containing generator according to claim 2, is characterized in that, described first-order nonlinear differential equation is discretized by using different integration methods to obtain the equation after discretization, include: 将在t时刻对所述一阶非线性微分方程乘以1-β得到的公式与在t+Δt时刻对所述一阶非线性微分方程乘以β得到的公式相加,得到由t时刻计算t+Δt时刻的计算式;Adding the formula obtained by multiplying the first-order nonlinear differential equation by 1-β at time t and the formula obtained by multiplying the first-order nonlinear differential equation by β at time t+Δt, the calculation is obtained at time t. Calculation formula at time t+Δt; 将所述在t时刻对所述一阶非线性微分方程乘以1-β得到的公式乘以
Figure FDA0003634755150000011
后,减去所述由t时刻计算t+Δt时刻的计算式,得到离散后的方程。
Multiply the formula obtained by multiplying the first-order nonlinear differential equation by 1-β at time t by
Figure FDA0003634755150000011
Then, subtract the calculation formula for calculating time t+Δt from time t to obtain a discrete equation.
4.根据权利要求1所述一种含发电机的电磁暂态仿真迭代求解方法,其特征在于,所述统一的非线性微分方程的计算式如下:4. a kind of electromagnetic transient simulation iterative solution method containing generator according to claim 1, is characterized in that, the calculation formula of described unified nonlinear differential equation is as follows:
Figure FDA0003634755150000012
Figure FDA0003634755150000012
式中,n为电路节点个数,nm为电机机械轴系质量块数,L(θ)为发电机电感矩阵,B为发电机绕组和节点的关联矩阵,J为发电机轴系转动惯性矩阵,D为阻尼矩阵,K为弹性系数矩阵,T为发电机机械转矩向量,KC为电容系数矩阵,KR为电阻系数矩阵,KL为电感系数矩阵,Ψ为节点磁链,θ为电机轴系转角。In the formula, n is the number of circuit nodes, nm is the number of mass blocks in the mechanical shaft system of the motor, L(θ) is the generator inductance matrix, B is the correlation matrix between the generator windings and nodes, and J is the rotational inertia matrix of the generator shaft system , D is the damping matrix, K is the elastic coefficient matrix, T is the generator mechanical torque vector, K C is the capacitance coefficient matrix, K R is the resistance coefficient matrix, K L is the inductance coefficient matrix, Ψ is the node flux linkage, θ is Motor shafting angle.
5.根据权利要求1所述一种含发电机的电磁暂态仿真迭代求解方法,其特征在于,所述利用牛顿迭代法对所述离散后的方程进行迭代求解,得到每个时间步长内的解,包括:5. A kind of electromagnetic transient simulation iterative solution method with generator according to claim 1, characterized in that, using Newton iteration method to iteratively solve the discrete equations, and obtain the value within each time step. solutions, including: 将所述离散后的方程转换为函数式,并获取函数式的雅克比矩阵;converting the discretized equation into a functional formula, and obtaining a Jacobian matrix of the functional formula; 将所述雅克比矩阵转换为可以利用牛顿迭代法的迭代式;transforming the Jacobian matrix into an iterative formula that can utilize Newton's iterative method; 利用牛顿迭代法对所述迭代式进行迭代求解,得到每个时间步长内的解。The iterative formula is iteratively solved by using the Newton iteration method, and the solution in each time step is obtained. 6.根据权利要求5所述一种含发电机的电磁暂态仿真迭代求解方法,其特征在于,所述利用牛顿迭代法对所述迭代式进行迭代求解,得到每个时间步长内的解,包括:6 . The iterative solution method for electromagnetic transient simulation with generator according to claim 5 , wherein the iterative solution of the iterative formula is performed by using the Newton iteration method, and the solution in each time step is obtained. 7 . ,include: 将‖f(xN+1,k+1)‖与设定的误差值进行比较,若小于所述设定的误差值则计算结束,否则按照牛顿迭代法的迭代式进行求解,xN+1,k+1为第k+1次迭代的估计值,k为迭代次数,x为状态量,f(xN+1,k+1)为第N+1步计算的第k+1次迭代的函数值,N为第N步计算。Compare ‖f(x N+1,k+1 )‖ with the set error value, if it is less than the set error value, the calculation ends, otherwise, solve according to the iterative formula of Newton iteration method, x N+ 1,k+1 is the estimated value of the k+1th iteration, k is the number of iterations, x is the state quantity, and f(x N+1,k+1 ) is the k+1th time calculated in the N+1th step The function value of the iteration, N is the calculation of the Nth step. 7.根据权利要求5所述一种含发电机的电磁暂态仿真迭代求解方法,其特征在于,所述利用牛顿迭代法的迭代式如下:7. a kind of electromagnetic transient simulation iterative solution method containing generator according to claim 5, is characterized in that, the described iterative formula utilizing Newton iteration method is as follows:
Figure FDA0003634755150000021
Figure FDA0003634755150000021
式中,xN+1,k为t+Δt时刻初始的估计值,xN+1,k+1为新的结果估计值,即下一次迭代初始的估计值。In the formula, x N+1, k is the initial estimated value at time t+Δt, and x N+1, k+1 is the new result estimated value, that is, the initial estimated value of the next iteration.
8.一种含发电机的电磁暂态仿真迭代求解系统,其特征在于,包括:8. An electromagnetic transient simulation iterative solution system containing a generator, characterized in that, comprising: 微分方程建立模块,用于基于发电机的电磁暂态仿真将发电机和电路系统建立统一的非线性微分方程;The differential equation building module is used to establish a unified nonlinear differential equation for the generator and the circuit system based on the electromagnetic transient simulation of the generator; 微分方程离散模块,用于采用积分方法对所述统一的非线性微分方程进行离散,得到离散后的方程;a differential equation discretization module, used for discretizing the unified nonlinear differential equation by using the integral method to obtain the discretized equation; 迭代求解模块,用于利用牛顿迭代法对所述离散后的方程进行迭代求解,得到每个时步内的解。The iterative solution module is used to iteratively solve the discrete equation by using the Newton iteration method to obtain the solution in each time step. 9.根据权利要求8所述一种含发电机的电磁暂态仿真迭代求解系统,其特征在于,所述微分方程离散模块,包括:9. The electromagnetic transient simulation iterative solution system containing a generator according to claim 8, wherein the differential equation discrete module comprises: 转换子模块,用于将所述统一的非线性微分方程转换为一阶非线性微分方程;a conversion submodule for converting the unified nonlinear differential equation into a first-order nonlinear differential equation; 离散子模块,用于利用不同的积分方法对所述一阶非线性微分方程进行离散得到离散后的方程。The discrete sub-module is used for discretizing the first-order nonlinear differential equation by using different integration methods to obtain a discretized equation. 10.根据权利要求9所述一种含发电机的电磁暂态仿真迭代求解系统,其特征在于,所述离散子模块,具体用于:10. A kind of electromagnetic transient simulation iterative solution system containing generator according to claim 9, is characterized in that, described discrete submodule is specifically used for: 将在t时刻对所述一阶非线性微分方程乘以1-β得到的公式与在t+Δt时刻对所述一阶非线性微分方程乘以β得到的公式相加,得到由t时刻计算t+Δt时刻的计算式;Adding the formula obtained by multiplying the first-order nonlinear differential equation by 1-β at time t and the formula obtained by multiplying the first-order nonlinear differential equation by β at time t+Δt, the calculation is obtained at time t. Calculation formula at time t+Δt; 将所述在t时刻对所述一阶非线性微分方程乘以1-β得到的公式乘以
Figure FDA0003634755150000031
后,减去所述由t时刻计算t+Δt时刻的计算式,得到离散后的方程。
Multiply the formula obtained by multiplying the first-order nonlinear differential equation by 1-β at time t by
Figure FDA0003634755150000031
Then, subtract the calculation formula for calculating time t+Δt from time t to obtain a discrete equation.
11.根据权利要求8所述一种含发电机的电磁暂态仿真迭代求解系统,其特征在于,所述统一微分方程的计算式如下:11. A kind of electromagnetic transient simulation iterative solution system containing generator according to claim 8, is characterized in that, the calculation formula of described unified differential equation is as follows:
Figure FDA0003634755150000032
Figure FDA0003634755150000032
式中,n为电路节点个数,nm为电机机械轴系质量块数,L(θ)为发电机电感矩阵,B为发电机绕组和节点的关联矩阵,J为发电机轴系转动惯性矩阵,D为阻尼矩阵,K为弹性系数矩阵,T为发电机机械转矩向量,KC为电容系数矩阵,KR为电阻系数矩阵,KL为电感系数矩阵,Ψ为节点磁链,θ为电机轴系转角。In the formula, n is the number of circuit nodes, nm is the number of mass blocks in the mechanical shaft system of the motor, L(θ) is the generator inductance matrix, B is the correlation matrix between the generator windings and nodes, and J is the rotational inertia matrix of the generator shaft system , D is the damping matrix, K is the elastic coefficient matrix, T is the generator mechanical torque vector, K C is the capacitance coefficient matrix, K R is the resistance coefficient matrix, K L is the inductance coefficient matrix, Ψ is the node flux linkage, θ is Motor shafting angle.
12.根据权利要求8所述一种含发电机的电磁暂态仿真迭代求解系统,其特征在于,所述迭代求解模块,包括:12. The electromagnetic transient simulation iterative solution system with generator according to claim 8, wherein the iterative solution module comprises: 函数式子模块,用于将所述离散后的方程转换为函数式,并获取函数式的雅克比矩阵;a functional submodule, used to convert the discrete equation into a functional formula, and obtain a Jacobian matrix of the functional formula; 迭代式子模块,用于将所述雅克比矩阵转换为可以利用牛顿迭代法的迭代式;an iterative submodule for converting the Jacobian matrix into an iterative formula that can utilize Newton's iterative method; 求解子模块,用于利用牛顿迭代法对所述迭代式进行迭代求解,得到每个时间步长内的解。The solving submodule is used to iteratively solve the iterative formula by using the Newton iterative method to obtain the solution in each time step. 13.根据权利要求12所述一种含发电机的电磁暂态仿真迭代求解系统,其特征在于,所述求解子模块具体用于:13. A kind of electromagnetic transient simulation iterative solution system containing generator according to claim 12, is characterized in that, described solution submodule is specifically used for: 将‖f(xN+1,k+1)‖与设定的误差值进行比较,若小于所述设定的误差值则计算结束,否则按照牛顿迭代法的迭代式进行求解,xN+1,k+1为第k+1次迭代的估计值,k为迭代次数,x为状态量,f(xN+1,k+1)为第N+1步计算的第k+1次迭代的函数值,N为第N步计算。Compare ‖f(x N+1,k+1 )‖ with the set error value, if it is less than the set error value, the calculation ends, otherwise, solve according to the iterative formula of Newton iteration method, x N+ 1,k+1 is the estimated value of the k+1th iteration, k is the number of iterations, x is the state quantity, and f(x N+1,k+1 ) is the k+1th time calculated in the N+1th step The function value of the iteration, N is the calculation of the Nth step. 14.根据权利要求12所述一种含发电机的电磁暂态仿真迭代求解系统,其特征在于,所述利用牛顿迭代法的迭代式如下:14. A kind of electromagnetic transient simulation iterative solution system including generator according to claim 12, is characterized in that, described iterative formula utilizing Newton iteration method is as follows:
Figure FDA0003634755150000041
Figure FDA0003634755150000041
式中,xN+1,k为t+Δt时刻初始的估计值,xN+1,k+1为新的结果估计值,即下一次迭代初始的估计值。In the formula, x N+1, k is the initial estimated value at time t+Δt, and x N+1, k+1 is the new result estimated value, that is, the initial estimated value of the next iteration.
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